COMPACTIFICATION OF STRATA OF ABELIAN DIFFERENTIALS

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1 COMPACTIFICATION OF STRATA OF ABELIAN DIFFERENTIALS MATT BAINBRIDGE, DAWEI CHEN, QUENTIN GENDRON, SAMUEL GRUSHEVSKY, AND MARTIN MÖLLER Abstract. We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne- Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve. Contents 1. Introduction 1 2. Comparison of strata compactifications Examples of the incidence variety compactification Plumbing and gluing differentials Flat geometric smoothing 47 References Introduction 1.1. Background. The Hodge bundle ΩM g is a complex vector bundle of rank g over the moduli space M g of genus g Riemann surfaces. A point (X, ω) ΩM g consists of a Riemann surface X of genus g and a (holomorphic) abelian differential ω on X. The complement of the zero section ΩM g is naturally stratified into strata ΩM g (µ) where the multiplicity of all the zeros of ω is prescribed by a partition µ = (m 1,..., m n ) of 2g 2. By scaling the differentials, C acts on ΩM g and preserves the stratification, hence one can consider the projectivized strata PΩM g (µ) in the projectivized Hodge bundle PΩM g = ΩM g/c. An abelian differential ω defines a flat metric with conical Date: April 4, Research of the first author is supported in part by the Simons Foundation grant # Research of the second author is supported in part by the National Science Foundation under the CAREER grant DMS and a Boston College Research Incentive Grant. Research of the third author was supported in part by ERC-StG Research of the fourth author is supported in part by the National Science Foundation under the grants DMS and DMS , and by a Simons Fellowship in Mathematics (Simons Foundation grant # to Samuel Grushevsky). Research of the fifth author is supported in part by ERC-StG

2 2 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER singularities such that the underlying Riemann surface X can be realized as a plane polygon whose edges are pairwise identified via translation. In this sense (X, ω) is called a flat surface or a translation surface. Varying the shape of flat surfaces induces a GL + 2 (R)-action on the strata of abelian differentials, called Teichmüller dynamics. A number of questions about surface geometry boil down to understanding the GL + 2 (R)- orbit closures in Teichmüller dynamics. What are their dimensions? Do they possess manifold structures? How can one calculate relevant dynamical invariants? From the viewpoint of algebraic geometry the orbit closures are of an independent interest for cycle class computations, which can provide crucial information for understanding the geometry of moduli spaces. Many of these questions can be better accessed if one can describe a geometrically meaningful compactification of the strata. In particular, the recent breakthrough of Eskin, Mirzakhani, Mohammadi [EM13; EMM15] and Filip [Fil16] shows that any orbit closure (under the standard topology) is a quasiprojective subvariety of a stratum. Thus describing the projective subvarieties that are closures of orbits in a compactified stratum can shed further light on the classification of orbit closures. Identify Riemann surfaces with smooth complex curves. The Deligne-Mumford compactification M g of M g parameterizes stable genus g curves that are (at worst) nodal curves with finite automorphism groups. The Hodge bundle ΩM g extends as a rank g complex vector bundle ΩM g over M g. The fiber of ΩM g over a nodal curve X parameterizes stable differentials that have (at worst) simple poles at the nodes of X with opposite residues on the two branches of a node. One way of compactifying PΩM g (µ) is by taking its closure in the projectivized Hodge bundle PΩM g over M g, and we call it the Hodge bundle compactification of the strata. Alternatively, one can lift a stratum PΩM g (µ) to the moduli space M g,n of genus g curves with n marked points by adding on each curve the data of the zeroes of differentials. Let M g,n be the Deligne-Mumford compactification of M g,n that parameterizes stable genus g curves with n marked points. Taking the closure of PΩM g (µ) in M g,n provides another compactification, which we call the Deligne-Mumford compactification of the strata. By combining the two viewpoints above, in this paper we describe a strata compactification that we call the incidence variety compactification PΩM inc g,n(µ). Let PΩM g,n be the projectivized Hodge bundle over M g,n, which parameterizes pointed stable differentials. Then the incidence variety compactification of PΩM g (µ) is defined as the closure of the stratum in the projectivization PΩM g,n. The incidence variety compactification records both the limit stable differentials and the limit positions of the zeros when abelian differentials become identically zero on some irreducible components of the nodal curve. It contains more information than the Hodge bundle compactification, because the latter loses the information about the limit positions of the zeros on the components of nodal curves where the stable differentials vanish identically. The incidence variety compactification also contains more information than the Deligne-Mumford compactification, because the latter loses the information on the relative sizes of flat surfaces corresponding to the components of nodal curves where the stable differentials are not identically zero.

3 COMPACTIFICATION OF STRATA 3 Our characterization of the boundary of the incidence variety compactification is in terms of a collection of (possibly meromorphic) differentials on the components of a pointed stable curve that satisfy certain combinatorial and residue conditions given by a full order on the vertices of the dual graph of the curve. Meromorphic differentials naturally arise in the description of the boundary objects, and the incidence variety compactification works just as well for the strata of meromorphic differentials, hence we take the meromorphic case into account from the beginning. In order to deal with meromorphic differentials, we consider the closure of the corresponding strata in the Hodge bundle over M g,n twisted by the polar part µ of the differentials, which we denote by KM g ( µ) and introduce in Section 2.5. Before we state the main result, let us first provide some motivation from several viewpoints for the reader to get a feel for the form of the answer that we get Motivation via complex analytic geometry. Given a pointed stable differential (X, ω, z 1,..., z n ) ΩM g,n, that is a stable curve X with marked points z 1,..., z n at the zeros of a stable differential ω, the question is whether it is the limit of a family of abelian differentials (X t, ω t ) contained in a given stratum ΩM g (µ) such that the z i are the limits of the zeros of ω t. Suppose f : X is a family of abelian differentials over a disk with parameter t, whose underlying curves degenerate to X at t = 0. If for an irreducible component X v of X the limit ω 0 := lim t 0 ω t is not identically zero, then on X v the limits of zeros of ω t are simply the zeros of ω. Thus our goal is to extract from this family a nonzero (possibly meromorphic) differential for every irreducible component of X where ω 0 is identically zero. The analytic way to do this is to take for every X v a suitable scaling parameter l v Z 0 such that the limit η v := lim t 0 t lv ω t Xv is well-defined and not identically zero. This is done in Lemma 4.1. Along this circle of ideas, we prove our main result by the plumbing techniques in Section Motivation via algebraic geometry. Now we sketch the algebro-geometric viewpoint of the above setting. Think of the family ω t as a section of the vector bundle f ω X / of abelian differentials on the fibers over the punctured disc, where ω X / is the relative dualizing line bundle. The Hodge bundle f ω X / extends f ω X / to a vector bundle over the entire disc, but so does any twisting f ω X / ( c v X v ) by an arbitrary integral linear combination of the irreducible components X v of the central fiber X. Based on the idea of Eisenbud-Harris limit linear series [EH86] (for curves of compact type), we want to choose coefficients c v in such a way that ω t extends over t = 0 to a section of the corresponding twisted dualizing line bundle, whose restriction η v to every irreducible component X v is not identically zero (see the discussion in [Che17] for more details). While the machinery of limit linear series for stable curves of arbitrary type is not available in full generality, our Definition 1.1 of twisted differentials works for all stable curves. It is modeled on the collection of η v defined above.

4 4 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER The next question is to determine which twisted differentials η = {η v } on a pointed stable curve X arise as actual limits of abelian differentials (X t, ω t ) that lie in a given stratum. First, η must have suitable zeros at the limit positions of the zeros of ω t. Moreover if X is reducible, the limit of canonical line bundles of X t is not unique, as it can be obtained by twisting the dualizing line bundle of X by any component X v (treated as a divisor in the universal curve). Accordingly the limit η of ω t on X can be regarded as a section of certain twisted dualizing line bundle, which is not identically zero on any component of X (for otherwise we can twist off such a component). Since the dualizing line bundle of X at a node is generated by differentials with simple poles, after twisting by c v X v, on one branch of the node the zero or pole order gains c v and on the other branch it loses c v, hence the zero and pole orders of η on the two branches of every node must add up to the original sum of vanishing orders 2 = ( 1) + ( 1). See [Che17, Section 4.1] for some examples and more details. We can then partially orient the dual graph Γ of X by orienting the edge from the zero of η to the pole, and leaving it unoriented if the differential has a simple pole at both branches. In this way, we obtain a partial order on the vertices v of Γ, with equality permitted (see also [FP16]). It turns out that a partial order is insufficient to characterize actual limits of abelian differentials in a given stratum. For the degenerating family (X t, ω t ), comparing the scaling parameters l v discussed in Section 1.2 extends this partial order to a full order on the vertices of Γ, again with equality permitted. The final ingredient of our answer is the global residue condition (4) in Definition 1.2 that requires the limit twisted differentials η to be compatible with the full order on Γ. Simply speaking, this global residue condition arises from applying Stokes formula to ω t on each level of Γ (i.e., truncating Γ at vertices that are equal in the full order), for t Motivation via flat surfaces. Degeneration of flat surfaces has been studied in connection with counting problems, for example in [EMZ03; EMR12; EKZ14]. Most of the degeneration arguments there rely on a theorem of Rafi [Raf07] on the comparison of flat and hyperbolic lengths for surfaces near the boundary. Rafi used a thick-thin decomposition of the flat surfaces by cutting along hyperbolically short curves. For each piece of the thick-thin decomposition Rafi defined a real number, the size, in terms of hyperbolic geometry. His main theorem says that after rescaling by size, hyperbolic and flat lengths on the thick pieces are comparable, up to universal constants. Rafi s size is closely related to the scaling parameters l v discussed above, implied by comparing our scaling limit (4.1) with the geometric compactification theorem [EKZ14, Theorem 10]. Rafi associated to every closed geodesic a flat representative of an annular neighborhood. Depending on the curvature of the boundary such a neighborhood is composed of flat cylinders in the middle and expanding annuli on both sides, any of the three possibly not being present. For degenerating families of flat surfaces this observation can be applied to the vanishing cycles of the family nearby a nodal fiber. In our result we use the vanishing cycles to read off the global residue condition that constrains degeneration of flat surfaces. While some of our terminology might be translated into the language of [Raf07] or [EKZ14], in the literature on flat surfaces and Teichmüller dynamics, no systematic

5 COMPACTIFICATION OF STRATA 5 attempt to describe the set of all possible limit objects under degeneration has been made. In Section 5 we provide an alternative proof of our main result by constructions of flat surfaces, where a pair of half-infinite cylinders correspond to two simple poles of a twisted differential attached together, and an expanding annulus that appears corresponds to a zero matching a higher order pole Level graphs. We now introduce the relevant notions that will allow us to state our result. Recall that Γ denotes the dual graph of a nodal curve X, whose vertices and edges correspond to irreducible components and nodes of X, respectively. First, we start with the idea of comparing irreducible components of X. A full order on the graph Γ is a relation on the set V of vertices of Γ that is reflexive, transitive, and such that for any v 1, v 2 V at least one of the statements v 1 v 2 or v 2 v 1 holds. We say that v 1 is of higher or equal level compared with v 2 if and only if v 1 v 2. We write v 1 v 2 if they are of the same level, that is if both v 1 v 2 and v 2 v 1 hold. We write v 1 v 2 if v 1 v 2 but v 2 v 1, and say that v 1 is of higher level than v 2. We call the set of maxima of V the top level vertices. We remark that equality is permitted in our definition of a full order. Any map l : V R assigning real numbers to vertices of Γ defines a full order on Γ by setting v 1 v 2 if and only if l(v 1 ) l(v 2 ). Conversely, every full order can be induced from such a level map, but not from a unique one. Later we will see that the levels are related to the scaling parameters introduced in Section 1.2, hence it would be convenient to consider the maps l : V R 0 assigning non-positive levels only, with l 1 (0) being the top level. We call a graph Γ equipped with a full order on its vertices a level graph, denoted by Γ. We will use the two notions full order and level graph interchangeably, and draw the level graphs by aligning horizontally vertices of the same level, so that a level map is given by the projection to the vertical axis, and the top level vertices are actually placed at the top (see the examples in Section 3) Twisted differentials. Throughout the paper we use ord q η to denote the zero or pole order of a differential η at q, and use Res q η to denote the residue of η at q. We now introduce the key notion of twisted differentials. Definition 1.1. For a tuple of integers µ = (m 1,..., m n ), a twisted differential of type µ on a stable n-pointed curve (X, z 1,..., z n ) is a collection of (possibly meromorphic) differentials η v on the irreducible components X v of X such that no η v is identically zero and the following properties hold: (0) (Vanishing as prescribed) Each differential η v is holomorphic and nonzero outside of the nodes and marked points of X v. Moreover, if a marked point z i lies on X v, then ord zi η v = m i. (1) (Matching orders) For any node of X that identifies q 1 X v1 with q 2 X v2, ord q1 η v1 + ord q2 η v2 = 2. (2) (Matching residues at simple poles) If at a node of X that identifies q 1 X v1 with q 2 X v2 the condition ord q1 η v1 = ord q2 η v2 = 1 holds, then Res q1 η v1 + Res q2 η v2 = 0.

6 6 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER These conditions imply that the set of zeros and poles of a twisted differential consists of the marked points z i and some of the nodes of the curve X Twisted differentials compatible with a level graph. We want to study under which conditions a twisted differential arises as a limit in a degenerating family of abelian differentials contained in a given stratum. As mentioned before, the conditions depend on a full order on the dual graph Γ of X, and we need a little more notation. Suppose that Γ is a level graph with the full order determined by a level function l. For a given level L we call the subgraph of Γ that consists of all vertices v with l(v) > L along with edges between them the graph above level L of Γ, and denote it by Γ >L. We similarly define the graph Γ L above or at level L, and the graph Γ =L at level L. Accordingly we denote by X >L the subcurve of X with dual graph Γ >L etc. For any node q connecting two irreducible components X v and X v, if v v, we denote by q + X v and q X v the two preimages of the node on the normalization of X. If v v, we still write q v ± for the preimages of the node, where the choice of which one is q v + is arbitrary and will be specified. In the same way we denote by v + (q) and v (q) the vertices at the two ends of an edge representing a node q. Definition 1.2. Let (X, z 1,..., z n ) be an n-pointed stable curve with a level graph Γ. A twisted differential η of type µ on X is called compatible with Γ if, in addition to conditions (0), (1), (2) in Definition 1.1, it also satisfies the following conditions: (3) (Partial order) If a node of X identifies q 1 X v1 with q 2 X v2, then v 1 v 2 if and only if ord q1 η v1 1. Moreover, v 1 v 2 if and only if ord q1 η v1 = 1. (4) (Global residue condition) For every level L and every connected component Y of X >L that does not contain a marked point with a prescribed pole (i.e., there is no z i Y with m i < 0) the following condition holds. Let q 1,..., q b denote the set of all nodes where Y intersects X =L. Then b j=1 Res q j η v (q j ) = 0, where we recall that q j X =L and v (q j ) Γ =L. We point out that a given twisted differential satisfying conditions (0), (1) and (2) may not be compatible with any level graph, or may be compatible with different level graphs with the same underlying dual graph. Condition (3) is equivalent to saying that, if v 1 is of higher level than v 2, then η v1 is holomorphic at every node of the intersections of X v1 and X v2, and moreover if v 1 v 2, then η v1 and η v2 have simple poles at every node where they intersect Main result. Recall that the incidence variety compactification of a stratum of abelian differentials PΩM g (µ) is defined as the closure of the stratum in the projectivized Hodge bundle over M g,n that parameterizes pointed stable differentials. For a stratum of meromorphic differentials, we take the closure in the twisted Hodge bundle by the polar part of the differentials (see Section 2 for details). Our main result characterizes boundary points of the incidence variety compactification for both cases.

7 COMPACTIFICATION OF STRATA 7 Theorem 1.3. A pointed stable differential (X, ω, z 1,..., z n ) is contained in the incidence variety compactification of a stratum PΩM g (µ) if and only if the following conditions hold: (i) There exists a level graph Γ on X such that its maxima are the irreducible components X v of X on which ω is not identically zero. (ii) There exists a twisted differential η of type µ on X, compatible with Γ. (iii) On every irreducible component X v where ω is not identically zero, η v = ω Xv. In Sections 1.2 and 1.3, we have briefly explained the ideas behind the necessity of these conditions. The sufficiency part is harder, that is, how can we deform pointed stable differentials satisfying the above conditions into the interior of the stratum? We provide two proofs, in Section 4 by using techniques of plumbing in complex analytic geometry, and in Section 5 by using constructions of flat surfaces. Recall that the incidence variety compactification combines the two approaches of compactifying the strata in the projectivized Hodge bundle PΩM g over M g and in the Deligne-Mumford space M g,n. In particular, it admits two projections π 1 to PΩM g and π 2 to M g,n by forgetting the marked points and forgetting the differentials, respectively. Hence our result completely determines the strata closures in the Hodge bundle compactification and in the Deligne-Mumford compactification. Corollary 1.4. A stable differential (X, ω) lies in the Hodge bundle compactification of a stratum if and only if there exists a pointed stable differential satisfying the conditions in Theorem 1.3 that maps to (X, ω) via π 1. A pointed stable curve (X, z 1,..., z n ) lies in the Deligne-Mumford compactification of a stratum if and only if there exists a pointed stable differential satisfying the conditions in Theorem 1.3 that maps to (X, z 1,..., z n ) via π 2. Remark 1.5. Given a level graph Γ, there can only exist finitely many twisted differentials η compatible with Γ and satisfying the conditions of the theorem, up to scaling η v on each irreducible component by a nonzero number. Indeed, if all the pole and zero orders of η at all nodes are given, it determines η uniquely up to scaling on each irreducible component. For any irreducible component X v on the bottom level, the zeros and poles of η v are prescribed outside of the nodes of X v, and at those nodes η v only has poles. As the total number of zeros and poles of η v, counted with multiplicity, is equal to 2g v 2, it implies that the sum of the orders of poles of η v at all nodes of X v is fixed, and hence there are finitely many choices. For each such choice, on every irreducible component intersecting X v the order of zero of η at any node where it intersects X v is thus uniquely determined. If another component on the bottom level intersects X v, then η has a simple pole at both branches of that node. Therefore, we can prove that there are finitely many choices of η up to scaling, by induction on the number of irreducible components of X. For later use we relate a twisted differential η and a pointed stable differential (X, ω, z 1,..., z n ) satisfying Theorem 1.3 as follows. Definition 1.6. Given a twisted differential η compatible with a level graph on X, define the associated pointed stable differential (X, ω, z 1,..., z n ) by taking ω to be equal

8 8 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER to η on all top level components and identically zero on components of lower levels, and by taking z 1,..., z n to be the set of zeros and poles of η away from the nodes of X. Conversely given a pointed stable differential (X, ω, z 1,..., z n ), if it is associated to a twisted differential η compatible with certain level graph on X, we say that η is an associated twisted differential of (X, ω, z 1,..., z n ). Using the above definition, we can restate Theorem 1.3 as that a pointed stable differential (X, ω, z 1,..., z n ) is contained in the incidence variety compactification of PΩM g (µ) if and only if there exists a level graph and a compatible twisted differential η of type µ such that (X, ω, z 1,..., z n ) is associated with η. Note that multiplying η by any nonzero numbers on components not of top level does not change the associated pointed stable differential. Hence given a pointed stable differential, the associated twisted differential η may not be unique. Indeed it is not unique even up to scaling on each irreducible component (see Example 3.2) History of the project and related work. Recently there have been several attempts via different viewpoints that aim at understanding the boundary behavior of the strata of abelian differentials. In a talk given in August 2008, Kontsevich discussed the problem of compactifying the strata, focusing on the matching order and matching residue conditions. In [Gen18] the third author studied the incidence variety compactification and applied the plumbing techniques to prove a special case of our main result when all the residues of a twisted differential are zero. In that case the global residue condition (4) obviously holds. Motivated by the theory of limit linear series, in [Che17] the second author studied the Deligne-Mumford strata compactification and deduced the necessity of the conditions (0), (1), and (2). He also obtained partial smoothing results in the case of curves of pseudo-compact type by combining techniques of algebraic geometry and flat geometry. In [FP16] Farkas and Pandharipande studied the Deligne-Mumford strata compactification by imposing the conditions (0), (1), (2), and (3), i.e., without the global residue condition (4). It turns out the corresponding loci are reducible in general, containing extra components of equal dimension or one less in the boundary of M g,n. Modulo a conjectural relation to Pixton s formula of the double ramification cycle, in [FP16, Appendix] Janda, Pandharipande, Pixton, and Zvonkine used the extra components to recursively compute the cycle classes of the strata in M g,n. In [MW17] Mirzakhani and Wright concentrated on a collapse of the Hodge bundle compactification by keeping track only of components where the stable differentials are not identically zero. They proved an identification between the tangent space of the boundary of a GL + 2 (R)-orbit closure and the intersection of the tangent space to the orbit closure with the tangent space to the boundary of their compactification. In [GKN17] Krichever, Norton, and the fourth author studied degenerations of meromorphic differentials with all periods real, where plumbing techniques are also used and a full order on the dual graph also arises. In Summer 2015 the authors of the current paper met in various combinations on several occasions, including in Bonn, Luminy, Salt Lake City, and Boston. After stimulating discussions, the crucial global residue condition and the proof of sufficiency emerged, finally completing the characterization of the compactification of the strata.

9 COMPACTIFICATION OF STRATA Applications. The main novel aspect of our determination of the closures of the strata is the global residue condition, which is used to characterize exactly those stable differentials that appear in the closure of a stratum in the Hodge bundle, and not to extraneous components. Thus any further work aimed at understanding the structure of strata compactifications must build on our description in an essential way. In particular, the global residue condition was a cornerstone in the work of Sauvaget [Sau17] who analyzed the boundary of the strata in the Hodge bundle in order to understand the homology classes of the strata closures, and in the work of Mullane [Mul17], who used the global residue condition to analyze certain divisor closures in M g,n, and discovered an infinite series of new extremal effective divisors. Furthermore, in [CC18] Qile Chen and the second author described algebraically the principal boundary of the strata in terms of twisted differentials, solving a problem that had been open for flat surfaces for more than a decade, while Ulirsch, Werner, and the fifth author applied in [MUW17] our compactification to solve the realizability problem for constructing the tropical Hodge bundle, posed in [LU17]. Finally in Section 3.6, as a consequence of our result we provide an efficient description for the degeneration of Weierstraß divisors on certain binary curve, recovering a main case in the work of [EM02] Organization of the paper. In Section 2 we review basic properties about moduli spaces of curves and abelian differentials. We further compare the incidence variety compactification to the Hodge bundle compactification and the Deligne-Mumford compactification, which illustrates what extra information the incidence variety compactification gains. In Section 3 we apply Theorem 1.3 to analyze explicitly a number of examples that characterize the significance and delicacy of the global residue condition. In Section 4 we prove Theorem 1.3 by the method of plumbing. Finally in Section 5 we provide an alternative proof by constructions of flat surfaces. In particular, Sections 3, 4 and 5 are independent. Depending on the reader s background and interests, they can be read in any order Acknowledgments. We are grateful to Curt McMullen for useful conversations on the proof of the main theorem, to Gavril Farkas and Rahul Pandharipande for communications with us at the Arbeitstagung at Max Planck Institute in Bonn, June 2015, and to Adrian Sauvaget and Dimitri Zvonkine for discussions at the Flat Surfaces workshop in Luminy, July We thank all the organizers and participants of the Flat Surfaces workshop for their interests in this problem and motivating us to work on the proof from the viewpoint of flat geometry. The second author thanks Alex Eskin, Joe Harris, and Anton Zorich for encouraging him over the years to study Teichmüller dynamics via algebraic geometry. The fourth author thanks Igor Krichever and Chaya Norton, with whom he has been developing related ideas and techniques for real-normalized meromorphic differentials. Finally we thank the anonymous referees for carefully reading the paper and many useful comments.

10 10 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER 2. Comparison of strata compactifications In this section we introduce the basic terminology about moduli spaces of curves and strata of abelian differentials. We also define the incidence variety compactification and compare it to the Hodge bundle and the Deligne-Mumford compactifications Moduli spaces of curves. Denote by M g the moduli space of curves of genus g that parameterizes smooth and connected complex curves of genus g, up to biholomorphism. Denote by M g,n the moduli space of n-pointed genus g curves that parameterizes smooth and connected complex curves of genus g together with n distinct (ordered) marked points. The space M g,n is a complex orbifold of dimension 3g 3 + n. Recall that a stable n-pointed curve is a connected curve with at worst nodal singularities, with n distinct marked smooth points, such that the automorphism group of the curve preserving the marked points is finite. Denote by M g,n the Deligne-Mumford compactification of M g,n parameterizing stable n-pointed genus g curves. Let S be a subgroup of the symmetric group S n. Then S acts on M g,n by permuting the marked points. The quotient of M g,n by S is denoted by M S g,{n} or simply M g,{n} when the group S is clear from the context. The dual graph of a nodal curve X is the graph Γ whose vertices correspond to the irreducible components of X. For every node of X joining two components v 1 and v 2 (possibly being the same component) the dual graph has an edge connecting v 1 and v 2. For every marked point there is a leg or equivalently a half-edge attached to the vertex corresponding to the irreducible component that contains the marked point. For a connected nodal curve X, if removing a node disconnects X, we say that it is a separating node. Otherwise we call it a non-separating node. If the two branches of a node belong to the same irreducible component of X, we say that it is an internal node Moduli spaces of abelian differentials. The moduli space of abelian differentials ΩM g is the complement of the zero section in the Hodge bundle ΩM g M g, which parameterizes pairs (X, ω) where X is a smooth and connected curve of genus g, and ω is a holomorphic differential on X. The space ΩM g has a natural stratification according to the orders of zeros of ω. Let µ = (m 1,..., m n ) be a partition of 2g 2 by positive integers. The stratum ΩM g (µ) of abelian differentials of type µ, as a subspace of ΩM g, parameterizes abelian differentials (X, ω) such that ω has n distinct zeros of orders m 1,..., m n, respectively. A stable differential on a nodal curve X is a (possibly meromorphic) differential ω on X which is holomorphic outside of the nodes of X and which has at worst simple poles at the nodes, with opposite residues. The Hodge bundle extends to a vector bundle ΩM g,n M g,n, the total space of the relative dualizing sheaf of the universal family f : X M g,n. The fiber of this vector bundle over a pointed nodal curve (X, z 1,..., z n ) parameterizes pointed stable differentials (X, ω, z 1,..., z n ), where ω is a stable differential on (X, z 1,..., z n ). There is a natural C -action on the Hodge bundle by scaling the differentials. This action preserves the stratification of ΩM g. The quotient of ΩM g under this action is

11 COMPACTIFICATION OF STRATA 11 denoted by PΩM g. In general, quotient spaces by such a C -action will be denoted by adding the letter P The incidence variety compactification. When abelian differentials degenerate, we want to keep track of the information about both the limit stable differentials and the limit positions of the marked zeros. This motivates the definition of the incidence variety. For a partition µ = (m 1,..., m n ) of 2g 2, the (ordered) incidence variety PΩM inc g,n(µ) is defined to be (2.1) PΩM inc g,n(µ) := { (X, ω, z 1,..., z n ) PΩM g,n : div (ω) = n m i z i }. The (ordered) incidence variety compactification PΩM inc g,n(µ) is defined to be the closure of the incidence variety inside PΩM g,n. If m i = m j for some i j, then interchanging z i and z j preserves the incidence variety. Hence the subgroup S of the symmetric group S n generated by such transpositions acts on PΩM inc g,n(µ) by permuting z 1,..., z n accordingly. We define the (unordered) incidence variety to be the quotient (2.2) PΩM inc g,{n} (µ) := PΩMinc g,n(µ)/s and define the (unordered) incidence variety compactification to be the quotient (2.3) PΩM inc g,{n}(µ) := PΩM inc g,n(µ)/s. Since the ordered and unordered incidence variety compactifications differ only by permuting the marked points, we call both of them the incidence variety compactification. In case we need to distinguish them, we do so by specifying n or {n} in the subscripts. On a smooth curve X, a non-zero differential ω determines its zeros z i along with their multiplicities. Hence PΩM inc g,{n} (µ) is isomorphic to the projectivized stratum of abelian differentials PΩM g (µ). In order to understand degeneration of abelian differentials in PΩM g (µ), we need to describe the boundary points of the incidence variety compactification Moduli spaces of pointed meromorphic differentials. The moduli spaces of meromorphic differentials from the viewpoint of flat geometry have been investigated by Boissy [Boi15a], including their dimensions and connected components. Let µ = (m 1,..., m r ; m r+1,..., m r+s ; m r+s+1,..., m r+s+l ) be an n-tuple of integers such that n i=1 m i = 2g 2, where m i > 0 for i r, m r+1 = = m r+s = 0, and m i < 0 for i > r + s. We call such µ a meromorphic type, and denote by ΩM g (µ) the moduli space of meromorphic differentials of type µ. It parameterizes n-pointed meromorphic differentials (X, ω, z 1,, z n ) on a smooth curve X such that the order of ω at z i is equal to m i, which may be a zero, regular point, or pole, corresponding to whether m i > 0, m i = 0, or m i < 0, respectively. As there are infinitely many meromorphic types for a fixed genus, these moduli spaces no i=1

12 12 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER longer form a stratification of a fixed ambient space, but by a slight abuse of language we still call them strata of meromorphic differentials The incidence variety compactification in the meromorphic case. To mimic the definition in the abelian case we need to generalize the notion of the Hodge bundle. We denote the polar part of µ by µ = (m r+s+1,..., m n ). We then define the pointed Hodge bundle twisted by µ to be the bundle n ) KM g,n ( µ) = f ω X /Mg,n ( m i Z i i=r+s+1 over M g,n, where we have denoted by Z i the image of the section of the universal family f given by the i th marked point. We call points (X, ω, z 1,..., z n ) KM g,n ( µ) pointed stable differentials (of type µ). Note that if µ is a holomorphic type, then the tuple µ is empty, so KM g ( µ) = ΩM g, and hence it recovers the preceding setting for the abelian case. For any meromorphic type µ, we perform the same operations inside the space KM g ( µ) as in the abelian case. The (ordered) incidence variety PΩM inc g,n(m 1,..., m n ) is defined as (2.4) PΩM inc g,n(µ) = { (X, ω, z 1,..., z n ) PKM g,n ( µ) : div (ω) = n m i z i }, and the (ordered) incidence variety compactification PΩM inc g,n(µ) is defined to be its closure in PKM g,n ( µ). We define the (unordered) incidence variety to be the quotient (2.5) PΩM inc g,{n} (µ) = PΩMinc g,n(µ)/s, where S is defined as in Section 2.3. Finally, we define the (unordered) incidence variety compactification to be the quotient (2.6) PΩM inc g,{n}(µ) = PΩM inc g,n(µ)/s. Again, by a slight abuse of language we sometimes skip the terms ordered and unordered, and refer to both of them as the incidence variety compactification of the strata of meromorphic differentials Comparison to the Hodge bundle and the Deligne-Mumford compactifications. As said earlier, the incidence variety compactification admits two forgetful maps to the projectivized Hodge bundle and to the Deligne-Mumford space, respectively. In this subsection, we discuss these maps and demonstrate that they indeed forget information. For simplicity of notation, we will only state this in the holomorphic case. The discussion can be easily generalized to the meromorphic case. Thus we start with a partition µ = (m 1,..., m n ) of 2g 2 by positive integers. The forgetful map (2.7) π 1 : PΩM inc g,{n}(µ) PΩM g (µ) forgets the marked points z 1,..., z n. More precisely, the image of (X, ω, z 1,..., z n ) under π 1 is (X, ω ), where X is obtained from X (as an unmarked curve) by blowing i=1

13 COMPACTIFICATION OF STRATA 13 down all P 1 tails and bridges, and ω can be identified with the restriction of ω to the remaining components (see [Gen18, Lemma 2.4] for details). The other forgetful map (2.8) π 2 : PΩM inc g,{n}(µ) M g,{n} forgets the stable differential ω, hence the image of (X, ω, z 1,..., z n ) under π 2 is just (X, z 1,..., z n ). Since a differential on a compact curve is determined uniquely (up to scaling) by the locations and orders of its zeros, both maps π 1 and π 2, when restricted to PΩM inc g,{n} (µ), are isomorphisms onto their respective images. However, over the boundary the fibers of both maps can be more complicated. In particular, they may no longer be finite, and neither image dominates the other. Proposition 2.1. For g 3 and n 2, the following properties hold: (i) The map π 1 is not finite, and there does not exist a map f : PΩM g (µ) M g,{n} such that f π 1 = π 2. (ii) The map π 2 is not finite, and there does not exist a map ( ) h : π 2 PΩM inc g,{n}(µ) PΩM g (µ) such that h π 2 = π 1. This proposition is not a priori clear. Indeed it uses the full strength of Theorem 1.3 about characterizing the boundary points of PΩM inc g,{n}(µ). As a result we see that the incidence variety compactification contains more information than both the Hodge bundle compactification and the Deligne-Mumford compactification of the strata. Proof. We first prove (i). Suppose X is the union of an elliptic curve X 1 with a curve X g 1 of genus g 1 intersecting at a node q + q such that (2g 4)q + K Xg 1 and that X 1 contains all the marked points z 1,..., z n. Put X g 1 on a higher level than X 1. The corresponding level graph Γ of X is represented on the left side of Figure 1. Take a stable differential ω on X such that ω Xg 1 is a holomorphic differential with a unique zero at q + of multiplicity 2g 4, and ω X1 is identically zero. Take a twisted differential η on X such that η Xg 1 = ω Xg 1 and such that η X1 is a meromorphic differential with div (η X1 ) = n i=1 m iz i (2g 2)q. One checks that η satisfies all the conditions in Definitions 1.1 and 1.2. In particular, the global residue condition follows from the residue theorem on X 1, because η X1 has a unique pole at q, and hence Res q η X1 = 0. Since η is compatible with Γ and (X, ω, z 1,..., z n ) is the associated pointed stable differential of η (see Definition 1.6), by Theorem 1.3 (X, ω, z 1,..., z n ) is contained in the incidence variety compactification PΩM inc g,{n}(µ). It implies that the π 1 -preimage of the stable differential (X, ω) is isomorphic to { (z 1,..., z n ) (X 1 ) n \ : n m i z i = (2g 2)q } /S, i=1

14 14 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER where is the big diagonal parameterizing the tuples where at least two marked points coincide. This preimage has dimension n 1. Since the π 2 -image retains the information about the positions of the marked points, this example implies that there does not exist a map f such that f π 1 = π 2. X g 1 q + X 1 q + 1 X g 2 q + 2 X 1 q q1 q2 X 2 Figure 1. The level graphs used in the proof of Proposition 2.1 Next we prove (ii). Let X be the union of two elliptic curves X 1, X 2 and a curve X g 2 of genus g 2, whose dual graph with a chosen full order Γ is represented on the right side of Figure 1. In this level graph X 1 and X g 2 are on the top level, both higher than X 2. Further suppose that (2g 6)q 2 + K X g 2 and that the points z 1,..., z n X 2 are chosen such that there exists a meromorphic differential η X2 on X 2 with div(η X2 ) = n i=1 m iz i 2q1 (2g 4)q 2 and such that Res q η X2 = Res 1 q η X2 = 0. 2 The existence of such η X2 is proved in [Boi15b]. Then (X, z 1,..., z n ) specifies a point in M g,{n}. Take a twisted differential η on X such that η X1 on X 1 is holomorphic and nowhere vanishing, η X2 is given as above on X 2, and η X2g 2 on X 2g 2 satisfies div(η X2g 2 ) = (2g 6)q 2 +. One checks that η satisfies all the required conditions to be compatible with Γ. In particular, the global residue condition holds because η X2 has zero residues at both q1 and q2. Hence by Theorem 1.3 the associated pointed stable differential (X, ω, z 1,..., z n ) is contained in the incidence variety compactification PΩM inc g,{n}(µ), where ω X1 = η X1, ω Xg 2 = η Xg 2, and ω is identically zero on X 2. Note that one can scale η on the top level components X 1 and X g 2 by a pair of nonzero scalars (λ 1, λ g 2 ), which does not affect its compatibility with Γ. The associated stable differential ω is scaled accordingly on X 1 and X g 2, but the underlying marked curve (X, z 1,..., z n ) remains the same. In other words, the π 2 -preimage of (X, z 1,..., z n ) in the incidence variety compactification contains the space of the projectivized pairs [λ 1, λ g 2 ], which is one-dimensional. Since the image of (X, ω, z 1,..., z n ) under π 1 retains the scaling information, there does not exist a map h such that h π 2 = π 1 holds. Remark 2.2. Conceptually speaking, the map π 2 fails to be injective for two reasons. First, for a component of X, there may exist a nontrivial linear equivalence relation between the marked points in that component. For example, let X be the union of a hyperelliptic curve Y of genus g 1, and a P 1 component, intersecting at two points q 1 and q 2 which are Weierstraß points of Y. Moreover suppose that all the marked points are contained in P 1. Then (X, z 1,..., z n ) can be the image of (X, ω, z 1,..., z n ) under π 2, where ω is identically zero on P 1 and restricts to Y as a differential with a zero of order 2k at q 1 and a zero of order 2(g k 2) at q 2, for any k {0,..., g 2}. The other

15 COMPACTIFICATION OF STRATA 15 reason is that some scaling factors for the differentials on the top level components are lost, as discussed in the second part of the proof of Proposition 2.1 (also see Lemma 2.3 below). When we analyze the examples presented in Section 3, the information of the dimension of fibers of π 2 will play a significant role. Hence we conclude this section by the following observation. Lemma 2.3. Let be an open boundary stratum of M g,n parameterizing nodal curves with a given dual graph. Let (X, z 1,..., z n ) be a curve in the intersection of the locus π 2 (PΩM inc g,n(µ)) with. Then the dimension of the fiber of π 2 over (X, z 1,..., z n ) is one less than the maximal number of connected components of the graph Γ =0, where the maximum is taken over all level graph structures Γ on Γ, such that there exists a compatible twisted differential of type µ on (X, z 1,..., z n ). Proof. Take a twisted differential η on X compatible with a chosen level graph Γ. Suppose (X, ω, z 1,..., z n ) is the pointed stable differential associated to (η, Γ). Then ω is equal to η restricted to all top level components, and is identically zero elsewhere. The image of (X, ω, z 1,..., z n ) under π 2 then only retains the information on the zeroes of ω on all the irreducible components of top level, and thus loses the information of the individual scaling factors of η on each such irreducible component. However, if two such top level irreducible components are connected by an edge, the matching residue condition at the corresponding node prescribes that the scale of η on these two components is equal, and thus there is only one scale parameter lost for each connected component of the graph of the top level components. The space of such scaling factors has projective dimension equal to the number of top level components minus one. The desired conclusion thus follows from applying this analysis to all possible level graphs and compatible twisted differentials. 3. Examples of the incidence variety compactification To illustrate all the aspects of the incidence variety compactification, in this section we study many examples. We start with examples discussing possible choices of level graphs and compatible twisted differentials for a fixed dual graph, thus showing that there are indeed choices involved, and that all our data are necessary. We then describe in detail the incidence variety compactification for a number of strata in low genus. Throughout this section, we use the following notation. Denote by K Xi the canonical line bundle of an irreducible component X i of a nodal curve X. A node joining two irreducible components X i and X j is denoted by q k. Moreover, if X i X j, then the node q k is obtained by identifying the points q + k X i with q k X j. We denote by Γ the dual graph of X, and by Γ a full order on Γ. For a twisted differential η and a stable differential ω on X, we use η i and ω i to denote their restrictions to the component X i, respectively. We also remind the reader to review Definition 1.6 for η and ω being associated with each other Cautionary examples. We present some examples that serve as an illustration for our formulation of the conditions on twisted differentials as well as a warning regarding the extent to which the choices of Γ and η determine each other.

16 16 BAINBRIDGE, CHEN, GENDRON, GRUSHEVSKY, AND MÖLLER Example 3.1. (Twisted differentials do not automatically satisfy the global residue condition) Let X be a curve with three components X 1 X 2 X 3 as represented in Figure 2. Suppose that X 1 and X 2 contain no marked poles, and suppose that g 3 = 0. Suppose that the only marked point on X 3 is a marked zero z 1, i.e. m 1 > 0. X 1 X 2 X 1 X 2 z 1 q 1 q 2 X 3 X 3 Figure 2. The curve and level graph used in Example 3.1 Let η be a twisted differential on X compatible with this level graph. Since X 3 is on the bottom level, η 3 has poles at q 1 and q 2. Let r 1 and r 2 be the residues of η 3 at q 1 and q 2, respectively. The residue theorem on X 3 = P 1 says that r 1 + r 2 = 0, with no further constraints. However, the global residue condition applied to the level of X 3 implies that r 1 = 0 and r 2 = 0, which does not follow from the relation r 1 + r 2 = 0. Example 3.2. (Non-uniqueness of associated twisted differentials) Theorem 1.3 says that a pointed stable differential lies in the incidence variety compactification of a given stratum if and only if there exists an associated twisted differential of the given type compatible with certain level graph (see Definition 1.6). However, such a twisted differential may not be unique (modulo scaling), even for a fixed level graph. For example, suppose X has three irreducible components X 1 X 2 X 3, where X 1 intersects X 2 at one point q 1, and X 2 and X 3 intersect at two points q 2 and q 3, and suppose all marked points z 1,..., z n lie on X 3, see Figure 3. X 3 X 1 q 1 q 2 q 3 z 1 z n X 2 X 1 X 2 X 3 Figure 3. The curve and level graph used in Example 3.2 Suppose η is a twisted differential compatible with the corresponding level graph Γ. Because X 1 has no marked points and it is on the top level, η 1 on X 1 is holomorphic, and it has a unique zero at q 1 + whose order is 2g 1 2. Moreover, η 3 on X 3 has all the prescribed zeros or poles at z i, hence the sum of its pole orders at q2 and q 3 is equal to 2g 2g 3. Suppose that η 3 has a pole of order k at q2 and a pole of order 2g 2g 3 k at q3. Then η 2 as a differential on X 2 has a pole of order 2g 1 at q1, and has zeros of orders k 2 and 2g 2g 3 k 2 at q 2 + and q+ 3, respectively. Finally applying the global residue condition to each level of Γ, respectively, it says that the sum of residues of η i on each irreducible component X i is zero, which follows from the residue theorem. Hence in this case the global residue condition imposes no further constraints on η.

17 COMPACTIFICATION OF STRATA 17 By the above analysis, there exists a twisted differential η compatible with Γ if and only if the pointed curve (X, z 1,..., z n ) satisfies K X1 (2g 1 2) q + 1, K X2 (k 2)q (2g 2g 3 k 2)q 3 + 2g 1q1, n K X3 m i z i kq2 (2g 2g 3 k)q3. i=1 For special curves X these conditions can be satisfied for different values of k. For instance, take all X i to be hyperelliptic curves of high genus, all q i ± to be Weierstraß points, and k to be even. In that case we obtain a number of distinct twisted differentials η, all of which are the same on X 1 (modulo scaling) but are different on X 2 and X 3 (even after modulo scaling). Suppose ω is the associated stable differential of η. Then ω 1 = η 1 is determined on the top level component X 1, which is the same for all the different choices of η (modulo scaling). Nevertheless, ω 2 and ω 3 are identically zero on the lower level components X 2 and X 3, respectively. Hence in this case different values of k in the above give rise to distinct twisted differentials η, but the associated stable differential (X, ω, z 1,..., z n ) remains to be the same. Example 3.3. (Pointed stable differentials do not determine the level graph) The previous example shows that there may be many different twisted differentials that are associated with a given pointed stable differential in the incidence variety compactification. We now show that a pointed stable differential (X, ω, z 1,..., z n ) does not necessarily determine a full order on the dual graph. The reason is that ω is identically zero on the lower level components of X, hence the lower level components may be ordered differently. An example illustrated in Figure 4 is given by a triangular subgraph on lower levels, attached to a top level component X 1. The only two marked points are z 1 X 3 and z 2 X 4. The two different level graphs are obtained by switching the ordering of X 3 and X 4 on the bottom two levels. X 1 X 1 X 1 X 2 q 2 q 1 q 3 X 3 z 1 z 2 X 4 X 2 X 2 q 4 X 4 X 3 X 3 X 4 Figure 4. The curve with two different level graphs used in Example 3.3